If a tire went flat and was run-on for any distance, it’s likely toast. They get really hot and really chewed up, really fast.
If you see a nail, caught it before pressure went too low, and plugged it, I don’t see the issue. Id personally only run it on the rear from that point though.
If the tires were sufficiently new, I’d want the right patch plug. I carry a plug kit in my two cars that don’t have spares. They also have a slime inflator kit just in case.
Like anything, this is a case by case matter. Often a nail puncture will hold air just fine and give time to do a legit repair. The only reason not to, IMO, is because you're swapping tires very soon.
I’m not convinced that a patch plug prevents corrosion better than a plug itself. I’m not seeing how it’s the case. Id consider any repaired tire to be compromised, regardless.
But that doesn’t mean you cant drive on it prudently for a long time.